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OF THE SQUARE ROOT.

Q. WHAT is a Square?

4. Any number multiplied by itself produces a square, Q. What is the Extraction of the Square Root?

A. If a square be given to find one side, it is called the extraction of the Square Root.

Q. How is the given Square to be prepared for extraction?

A. By pointing off at every two figures, from the units place, both ways for a resolvend.

Q. What is a Surd?

A. It is an imperfect Square, or such a number whose Square Root can never be exactly found.

EXAMPLES.

1 What is the Square of 17.1? 2 What is the Square of .09 ? 3 What is the Square of .0094? 4 What is the Square Root of 4712.81261?

5 What is the Square Root of 9712.7180512

6 What is the Square Root of 3,1721812 ?

7 What is the Square Root of 1.3976121?

8 What is the Square Root of 761.801216?

9 What is the Square Root of 0007612816?

10 What is the Square Root

of 4.000067121 ?

Answ. 292. 41
Answ. .0081
Answ..00008836

Answ. 68.649 +

Answ. 98.553+

Answ. 1.78106+

Answ. 1.1822+

Answ. 27.6007+

Answ. .02759+

Answ. 2.000016+

11 There is an army consisting of a certain number of men, who are placed rank and file, that is, in the form of a Square, each side having 472 men; I demand how many men the whole square contains? Aus. 222784 men,

12 The floor of a certain great room is made exactly square, each side of which contains 75 feet? I demand how many Square feet are contained therein? Ans. 5625 feet.

13. Suppose 12544 Soldiers are to be put into rank and file, in the form of an equal square; Idemand how many soldiers will be in the front,and how many deep? Ans 112.

14 A certain square pavement contains 197136 square stones ail of the same size; I demand how many are contained in one of its sides? Ans. 414.

15 The wall of a town is 17 feet high, which is surrounded by a moat, of 20 feet in breadth; I demand the length of a ladder which shall reach from the outside of the moat to the top of the wall? Answer 26.2+feet.

OF THE SQUARE ROOT OF A. VULGAR FRAÇTION.

Q. How is the Square Root of a Vulgar Fraction extracted?

A. Reduce the Fraction to its lowest terms.

2 Extract the Square Root of the Numerator for a new Numerator, and the Square Root of the Denominator for a new Denominator.

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3 If the fraction be a Surd, reduce it to a Decimal, and then extract the Square Root from it.

4 The Decimal Fractions must consist of an eyen number of places, as two, four, &c.

EXAMPLES.

1 What is the Square Root of 3044
2 What is the Square-Root of 3450 ?
3 What is the Square-Root of 308

Surds.

8192

4 What is the Square-Root of 3168? 5 What is the Square-Root of 298? 6 What is the Square-Root of 337?

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OF THE SQUARE-ROOT OF A MIXT NUMBER.

Q. How is the Square-Root of a mixt number extracted? A. Reduce the Fractional part of the mixt number to its lowest term.

2 Reduce the mixt number to an improper fraction. 3 Extract the Roots of the Numerator and Denominator, for a new Numerator and Denominator.

4 If the mixt number given be a Surd, reduce the frac tional part to a Decimal, and annex it to the whole number, and extract the Square-Root from the whole.

EXAMPLES.

1 What is the Square-Root of 3738? 2 What is the Square-Root of 17? 3 What is the Square-Root of 5288 SURDS.

4 What is the Square-Root of 7614? 5 What is the Square-Boot of 7,2?

Answer 6
Answer 51
Answer 23

Answer 8.7649+
Answer 2.7961+

Q.

OF THE CUBE-ROOT:

WHAT is a Cube?

A. Any Number multiplied by its Square, produces a Cube.

Q. What is the extraction of the Cube-Root ?

A. If a Cube be given to find out a Number, which, being multiplied into its Square, produceth the number given; this is called the extraction of the Cube-Root

Q. How is the given Cube to be prepared for Extraction? A. By pointing off at every three Figures, both ways, from the unit's place, for a resolvend.

Q. What is a Surd?

A. It is an imperfect Cube, or such a number, whose Cube-Root can never be exactly found.

Q. What is the rule for extracting the Cube-Root of a Number:

A. This the first figure sought is the Root of the greatest Cube contained in the first member, and it is called a; then 3aa3+ a is the Divisor, which finds a new figure called e; then 3aae +eca+eee is the Subtrahend or Number to be subducted; which operation is to be continued to every resolvend.

Note. This rule being somewhat dark, I shall, by way of illustration, subjoin the operation at large for extracting the Cube-Root of any number.

What is the Cube-Root of 444194.947 ?

(1) Let the given Number be pointed as before directed;

thus: 444194.947

(2) The first member, which contains the greatest Cube, is 444; and the nearest Root, whose Cube is not greater than it, is 7, which set.

thus: 444194.947(7

(3) The Cube of 7 is 343, which set down and subtract, annexing the next three figures, or member, viz. 194 for a resolvend;

thus: 444194.947(7

343

101194 Resolvend.

(4) The number 7, in the Root is called a; then by the Rule 3aa3a is the Divisor; thus,

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Divisor 1491 Saa+3a

(5) The next figure in the Root, viz. 6 (found by common Division) is called e; then by the rule 3aae+3eea+eee is the Subtrahend, or Number to be subducted; thus,

147=3αα
6=e

882-Saae

756=3eea

216 eee

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Sub. 959763aae +3eea+eee

444194.947 (76
343

1491) 101194 Resolvend
95976 Subtrahend

5218.947 Resolvend.

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(6) When the next number is brought down, viz. 947 as before, both figures in the Root, viz. 76 must be called a; then to find a Divisor to this last Resolvend, say, as before, 3aa+3a; thus,

76=a

76=

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